Question 12
Equilateral $\triangle ABC$ is inscribed in a circle of radius $2.$ Extend $\overline{AB}$ through $B$ to point $D$ so that $AD = 13,$ and extend $\overline{AC}$ through $C$ to point $E$ so that $AE = 11.$ Through $D,$ draw a line $\ell_1$ parallel to $\overline{AE},$ and through $E,$ draw a line $\ell_2$ parallel to $\overline{AD}.$ Let $F$ be the intersection of $\ell_1$ and $\ell_2.$ Let $G$ be the point on the circle that is collinear with $A$ and $F$ and distinct from $A.$ Given that the area of $\triangle CBG$ can be expressed in the form $\frac{p\sqrt{q}}{r},$ where $p,$ $q,$ and $r$ are positive integers, $p$ and $r$ are relatively prime, and $q$ is not divisible by the square of any prime, find $p + q + r.$